Event Studies
Simple Panel Data Approaches with Binary Treatment
INTRODUCTION
Learning Outcomes
Textbook References
- First lecture: none
- After covering time series event studies: Chapter 17 in Huntington-Klein (2025): https://www.theeffectbook.net/ch-EventStudies.html
Further Reading
Reference articles:
Latter two articles more advanced and modern — how people do it these days.
Also see NBER 2023 Methods Lectures on Linear Panel Event Studies: https://www.nber.org/conferences/si-2023-methods-lectures-linear-panel-event-studies
TWO PERIODS
SETTING
Simple Event Study Setting
Begin with the simplest possible panel setting with binary treatment:
- Two periods with \(N\) units:
- No treatment in period 1.
- All units treated in period 2.
- Data: outcomes \((Y_{i1}, Y_{i2})\).
Object of interest: “average effect of treatment”
Simple Estimator — Average Change
Simplest approach: compute average change in \(Y_{it}\) across periods \[ \widehat{AE}_{ES} = \dfrac{1}{N}\sum_{i=1}^N (Y_{i2}- Y_{i1}). \tag{1}\]
Estimator (1) — simplest example of event study estimators (see Freyaldenhoven et al. 2021; Miller 2023).
Example Framework
Possible empirical framework
- Units \(i\): firms that make phones
- Outcome \(Y_{it}\): their stock price
- Periods:
- One week before Apple announces the iPhone.
- One week after the announcement
Effect of interest: change in stock prices due to the announcement of iPhone
What Does (1) Do?
Proposition 1 (Asymptotics for \(\widehat{AE}_{ES}\)) Let
- (Cross-sectional random sampling): \((Y_{i1}, Y_{i2})\) be independent and identically distributed (IID)
- Finite first moments: \(\E[\abs{Y_{it}}]<\infty\)
Then \[ \widehat{AE}_{ES} \xrightarrow{p} \E[Y_{i2} - Y_{i1}]. \]
CAUSAL ANALYSIS
Causal Framework
Is \(\E[Y_{i2} - Y_{i1}]\) interesting (=causal)?
Need a causal framework to talk about causal effects!
Work in the familiar potential outcomes framework:
- \(Y_{it}^0\) — outcome for \(i\) in period \(t\) if not treated
- \(Y_{i1}^1\) — outcome for \(i\) in period \(t\) if treated
- Treatment effect for \(i\) in \(t\): \(Y_{it}^1- Y_{it}^0\)
For short, use \(Y_{it}^d\) where \(d=0, 1\)
Limit of ES Estimator and Causality
Potential and realized outcomes are connected as \[ Y_{i2} = Y_{i2}^1, \quad Y_{i1} = Y_{i1}^0. \]
It follows that \[ \widehat{AE}_{ES} \xrightarrow{p} \E[Y_{i2}^2- Y_{i1}^1]. \]
\(\E[Y_{i2}^2- Y_{i1}^1]\) is not necessarily a treatment effect — mixes effect of treatment and effects of time!
Example
Context Again consider the iPhone example. Then
- \(Y_{i2}^1 - Y_{i2}^0\) — treatment effect, change in price because of the iPhone announcement
- \(Y_{i2}^0 - Y_{i0}^0\) — change in a world without iPhone
We see combination of both changes \[ Y_{i2} - Y_{i1} = Y_{i2}^1- Y_{i1}^0 = [Y_{i2}^1- Y_{i2}^0] + [Y_{i2}^0 - Y_{i0}^0] \]
Solution: Restrict Changes over Time
Simple solution: rule out changes over time
Assumption: no variation in potential outcomes \[ Y_{i2}^d= Y_{i1}^d, \quad d=0, 1 \]
Then \(\widehat{AE}_{ES}\) is estimating a causal parameter — average effects \[ \begin{aligned} \widehat{AE}_{ES} & \xrightarrow{p} \E[Y_{i1}^1- Y_{i1}^0] = \E[Y_{i2}^1- Y_{i2}^0] \end{aligned} \]
Relaxing Time Invariance: No Trends
Time invariance very strict. Why use it if we only work with averages?
Weaker assumption:
Assumption (no trends):
\[
\E[Y_{i2}^d] = \E[Y_{i1}^d], \quad d=0, 1
\]
Allows random variation in potential outcomes between times
Summary so Far
Proposition 2 (Causal asymptotics for \(\widehat{AE}_{ES}\)) Let
- Assumptions of Proposition 1 hold
- No trends assumption hold
Then \(\widehat{AE}_{ES}\) consistent for causal parameters: \[ \widehat{AE}_{ES} \xrightarrow{p} \E[Y_{i1}^1- Y_{i1}^0] = \E[Y_{i2}^1- Y_{i2}^0] \]
REGRESSION INTERPRETATION
Regression Setting
Can also connect \(\widehat{AE}_{ES}\) and OLS
Consider regression model \[ \begin{aligned} Y_{it} & = \beta_0 + \beta_1 D_{it} + u_{it}, \\ D_{it} & = \begin{cases} 1, & t= 1 \\ 0, & t =0 \end{cases} \end{aligned} \tag{2}\] where we simply treat \((Y_{i1}, D_{i1})\) and \((Y_{i2}, D_{i2})\) as separate observations
Event Study and OLS
Proposition 3 (\(\widehat{AE}_{ES}\) is OLS) For \(\beta_1\) of regression (2) \[ \widehat{AE}_{ES} = \hat{\beta}_1^{OLS} \]
- Can use all results developed for OLS for \(\widehat{AE}_{ES}\)
- Regressing \(Y_{it}\) on \(D_{it}\) gives a causal parameter
- Under no trends
- No experiment was necessary
Event Study and Regression
A way to think about regression in causal settings:
Write down the regression in terms of parameters of interest: e.g. let \[ \beta_0 = \E[Y_{i0}^0], \quad \beta_1 = \E[Y_{i2}^1- Y_{i2}^0] \]
Connect regression to potential outcomes: what is \(u_{it}\) in terms of potential outcomes?
Check properties of this \(u_{it}\). If \(u_{it}\) is “nice”, apply OLS (or another method)
MULTIPLE PERIODS
ESTIMATION AND CAUSAL FRAMEWORK
Multiple Period Framework
- Often have more than 2 periods of data
- Want to use that data
New framework:
- \(T\) periods in total
- Treatment starts in period \(t_0\)
- We see \(Y_{it}^0\) for \(t<t_0\) and \(Y_{it}^1\) for \(t\geq t_0\)
Expanded Regression
New variables for treatment: \[ D_{it, \tau} = \begin{cases} 1, & t= \tau, \\ 0, & t\neq \tau \end{cases} \]
Can try similar regression: \[ Y_{it} = \beta_0 + \sum_{\tau = t_0}^{T} \beta_\tau D_{it, \tau} + u_{it}. \tag{3}\]
Estimates
Fairly easy to show that \[ \begin{aligned} \hat{\beta}^{OLS}_{\tau} & = \dfrac{1}{N} \sum_{i=1}^N Y_{i\tau} - \dfrac{1}{N(t_0-1)} \sum_{i=1}^N\left[ Y_{i1} + \dots + Y_{it_0-1} \right] \\ & \xrightarrow{p} \E\left[Y_{i\tau}^1 - \dfrac{1}{t_0-1}(Y_{i1}^0+ \dots + Y_{it_0-1}^0) \right] \end{aligned} \] More general version of the simple estimator of before
Dynamic Treatment Effects?
If \(\beta_{\tau}\) — average effect in period \(\tau\), then model (3) seems to allow for dynamic effects
Dynamic effects often realistic: effect of treatment may grow or disappear over time. Example: impact of job training on earnings:
- Disappearing: you forget the training over time
- Increasing: job training lets you jump to a higher position and gain experience quicker for the rest of your life
No Trends Rule Out Dynamics
Suppose that the no trends assumption holds \[ \begin{aligned} & \E\left[Y_{i\tau}^1 - \dfrac{1}{t_0-1}(Y_{i1}^0+ \dots + Y_{it_0-1}^0) \right] = \E[Y_{it}^1 - Y_{it}^0] \end{aligned} \] where right hand side does not depend on \(t\)
The no trends assumption rules out dynamic treatment effects
This is bad, since we want to allow for dynamics
Relaxing No Trends Assumption
Can relax no trends to only affect one of the potential outcomes — makes more sense to restrict the untreated outcome (why?)
Assumption (no trends in the baseline): \(\E[Y_{it}^0]\) does not depend on \(t\)
Estimation of Dynamic Average Treatment Effects
Under the no trends in the baseline assumption: \[ \begin{aligned} & \E\left[Y_{i\tau}^1 - \dfrac{1}{t_01}(Y_{i1}^0+ \dots + Y_{it_0-1}^0) \right] = \E[Y_{i \tau}^1 - Y_{i\tau}^0] \end{aligned} \] Right hand side is average effect in period \(\tau\)
ASYMPTOTIC PROPERTIES
Consistency in the Multivariate Case
Proposition 4 Let
- Assumptions of Proposition 1 hold
- No trends in the baseline assumption hold
Then \(\hat{\beta}_{\tau}\) consistently estimates the average treatment effect in period \(\tau\) given by \(\E[Y_{i \tau}^1 - Y_{i\tau}^0]\).
Unbiasedness of OLS
Moreover, under no trends in the baseline \[ \E[\hat{\beta}_{\tau}] = \E[Y_{i \tau}^1 - Y_{i\tau}^0] \] In other words, the OLS estimator is unbiased
Asymptotic Distribution
Proposition 5 Let
- No trends in the baseline assumption hold
- \((Y_{i1}, Y_{i2}, \dots, Y_{iT})\) be IID (over \(i\))
- Finite second moments: \(\E[Y_{it}^2]<\infty\)
Then there exists some variance \(V\) such that \[ \sqrt{N}\left( \hat{\beta}_{\tau} - \E[Y_{i \tau}^1 - Y_{i\tau}^0] \right) \Rightarrow N(0, V) \]
Asymptotic Variance I
Need variance \(V\) to construct confidence intervals for \(\E[Y_{i \tau}^1 - Y_{i\tau}^0]\). How to find \(V\)? Remember the CLT:
If \(X_1, X_2, \dots\) are IID random variables with \(\E[X_i]=\mu\) and \(\E[X^2]<\infty\), then \[ \sqrt{N}\left( \dfrac{1}{N}\sum_{i=1}^N X_i - \mu \right)\Rightarrow N\left(0, \var(X_i) \right) \] Asymptotic variance = variance of \(X_i\)
Asymptotic Variance II
Can apply this idea to find \(V\). Can write \[ \begin{aligned} \hat{\beta}^{OLS}_{\tau} & = \dfrac{1}{N} \sum_{i=1}^N Z_i\\ Z_i & = Y_{i\tau} - \dfrac{1}{(t_0-1)}\left[ Y_{i1} + \dots + Y_{it_0-1} \right] \end{aligned} \]
Then we get that \[ V = \var(Z_i) \]
Estimator for Asymptotic Variance
Can estimate \(V\) with \[ \hat{V} = \widehat{\var}(Z_i) = \dfrac{1}{N}\left(Z_i - \dfrac{1}{N}\sum_{j=1}^N Z_j \right)^2 \]
Estimated standard error of \(\hat{\beta}_{\tau}\): \[ \widehat{se}(\hat{\beta}_{\tau}) = \sqrt{ \dfrac{\hat{V}}{N} } \]
Inference on Average Effects I
Can now construct confidence intervals and hypothesis tests about \(\E[Y_{i \tau}^1 - Y_{i\tau}^0]\). E.g. an asymptotic 95% confidece interval: \[ \widehat{CI}_{95\%} = \left[\hat{\beta}^{OLS}_{\tau} - z_{1-\alpha/2}\widehat{se}(\hat{\beta}_{\tau}), \hat{\beta}^{OLS}_{\tau} + z_{1-\alpha/2}\widehat{se}(\hat{\beta}_{\tau}) \right] \] where the critical values \(z_{1-\alpha/2}\) come from the standard normal distribution \[ z_{1-\alpha/2}= \Phi^{-1}\left(1 - \dfrac{\alpha}{2} \right) \]
Inference on Average Effects II
But what if we want to set the joint hypothesis \[ H_0: \beta_{\tau} = 0, \quad \tau = t_0, \dots, T \]
- This \(H_0\) is a hypothesis above a vector of effects
- Need a joint asymptotic distribution to write down a test
Joint Asymptotic Distribution
Proposition 6 (Joint Asymptotics for Estimated Effects) Let
- No trends in the baseline assumption hold
- \((Y_{i1}, Y_{i2}, \dots, Y_{iT})\) be IID (over \(i\))
- Finite second moments: \(\E[Y_{it}^2]<\infty\)
Then there exists some variance-covariance matrix \(\bV\) such that \[ \sqrt{N}(\hat{\bbeta} -\bbeta) \Rightarrow N(0, \bV) \]
Joint Inference on Average Effects
Can use Proposition 6 to create a Wald test
EMPIRICAL ILLUSTRATION
Slide
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References
Panel Data I: Event Studies